arXiv: Representation Theory | 2019
A criterion for discrete branching laws for Klein four symmetric pairs and its application to $E_{6(-14)}$
Abstract
Let $G$ be a noncompact connected simple Lie group, and $(G,G^\\Gamma)$ a Klein four symmetric pair. In this paper, the author shows a necessary condition for the discrete decomposability of unitarizable simple $(\\mathfrak{g},K)$-modules for Klein for symmetric pairs. Precisely, if certain conditions hold for $(G,G^\\Gamma)$, there does not exist any unitarizable simple $(\\mathfrak{g},K)$-module that is discretely decomposable as a $(\\mathfrak{g}^\\Gamma,K^\\Gamma)$-module. As an application, for $G=\\mathrm{E}_{6(-14)}$, the author obtains a complete classification of Klein four symmetric pairs $(G,G^\\Gamma)$ with $G^\\Gamma$ noncompact, such that there exists at least one nontrivial unitarizable simple $(\\mathfrak{g},K)$-module that is discretely decomposable as a $(\\mathfrak{g}^\\Gamma,K^\\Gamma)$-module and is also discretely decomposable as a $(\\mathfrak{g}^\\sigma,K^\\sigma)$-module for some nonidentity element $\\sigma\\in\\Gamma$.